Showing content from https://dlmf.nist.gov/4.39 below:
§4.39 Continued Fractions ⣠Hyperbolic Functions ⣠Chapter 4 Elementary Functions
§4.39 Continued Fractions â
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Keywords:
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continued fractions, hyperbolic functions, inverse hyperbolic functions
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Permalink:
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http://dlmf.nist.gov/4.39
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Annotations for Ch.4
4.39.1 tanh â¡ z = z 1 + z2 3 + z2 5 + z2 7 + ⯠, z â ± 1 2 â¢ Ï â¢ i , ± 3 2 â¢ Ï â¢ i , ⦠. â
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Symbols:
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Ï : the ratio of the circumference of a circle to its diameter, tanh â¡ z : hyperbolic tangent function, i : imaginary unit and z : complex variable
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A&S Ref:
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4.5.70
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http://dlmf.nist.gov/4.39.E1
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Annotations for §4.39 and Ch.4
4.39.2 arcsinh â¡ z 1 + z2 = z 1 + 1 â
2 ⢠z2 3 + 1 â
2 ⢠z2 5 + 3 â
4 ⢠z2 7 + 3 â
4 ⢠z2 9 + ⯠, â
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Symbols:
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arcsinh â¡ z : inverse hyperbolic sine function and z : complex variable
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A&S Ref:
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4.6.36
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Permalink:
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http://dlmf.nist.gov/4.39.E2
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Annotations for §4.39 and Ch.4
where z is in the open cut plane of Figure 4.37.1(i).
4.39.3 arctanh â¡ z = z 1 â z2 3 â 4 ⢠z2 5 â 9 ⢠z2 7 â ⯠, â
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Symbols:
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arctanh â¡ z : inverse hyperbolic tangent function and z : complex variable
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A&S Ref:
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4.6.35
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Permalink:
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http://dlmf.nist.gov/4.39.E3
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Annotations for §4.39 and Ch.4
where z is in the open cut plane of Figure 4.37.1(iii).
For these and other continued fractions involving inverse hyperbolic functions see Lorentzen and Waadeland (1992, pp. 569â571). See also Cuyt et al. (2008, pp. 211â217).
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